47 votes

Why isn't the Weierstrass function $\sum_{n=0}^\infty a^n \cos(b^n\pi x)$ differentiable?

Nothing is preventing us from taking the derivative of any finite partial sum of this series. This is a trigonometric polynomial and it has derivatives of all orders. However, this infinite sum ...
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31 votes

Does there exist a function which converts exponentiation into addition?

As Stinking Bishop mentions in the comments, if $f(a)+f(b)=f(a^b)$, then it follows (by interchanging $a$ and $b$) that $f(b)+f(a)=f(b^a)$. Hence, $f(a^b)=f(b^a)$ for all $a,b\in\mathbb R^+$. Setting $...
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25 votes

Why isn't the Weierstrass function $\sum_{n=0}^\infty a^n \cos(b^n\pi x)$ differentiable?

As other answers have pointed out, while the partial sums of the function converge, the partial sums of the derivatives do not. To see this, I just did a quick calculation. Let $f_m(x)$ be the partial ...
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  • 1,081
21 votes
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How does this definition of a computable function work?

Let's consider $f(n) = \frac{7n}5$. Whatever “computable” means, this function should have that property. For example, if we put in $n=2$ the computer can print out ...
18 votes
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How to show analytically that $x^4-4x^3-2x^2+16x+24=0$ has no solutions in $\mathbb{R}$?

$$f(x)=x^4-4x^3-2x^2+16x+24=(x^2-2x-4)^2+2x^2+8>0.\tag{1}$$ My motivation: the terms $-4x^3$ and $16x$ are annoying since they can be either positive or negative, so I wanted to put them in a ...
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  • 8,589
16 votes

Why isn't the Weierstrass function $\sum_{n=0}^\infty a^n \cos(b^n\pi x)$ differentiable?

It's not that it is an infinite sum, any analytic function can be written as an infinite sum by definition and is differentiable. The issue is that the function has the nature of a fractal, at ...
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  • 14.9k
14 votes

Why isn't the Weierstrass function $\sum_{n=0}^\infty a^n \cos(b^n\pi x)$ differentiable?

In order to calculate the derivative of a function at a given point (e.g. $x=0$), you just need to keep zooming on this point until the curve gets smooth: Oh, wait...
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  • 3,206
13 votes
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How can $f(x) = x^2$ be continuous?

You're mixing up a lot of concepts there, and that's part of what's confusing you. To give you some direction on things: The size of sets For finite sets, it's easy to say that their size is the ...
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13 votes
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When $f(z) = 1/z$ and $g(z) = 1-z$, why is $f \circ g \circ f \circ g \circ f \circ g (z) = g \circ f \circ g \circ f \circ g \circ f (z) = z$?

The functions $f(z) = 1/z$ and $g(z) = 1 - z$ are both linear fractional transformations; that is, they have the form $\frac{az + b}{cz + d}$ for suitable complex numbers $a, b, c, d$ such that $ad - ...
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12 votes

Finding $\displaystyle \sum \limits _{n=1}^{\infty}\frac{(-1)^n (H_{2n}-H_{n})}{n(2)^n \binom{2n}{n}}$

Clear["Global`*"] f[n_] := (-1)^ n (HarmonicNumber[2 n] - HarmonicNumber[n])/(n*2^n*Binomial[2 n, n]) The terms of the sum converge to zero rapidly <...
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11 votes

Proving $\ddot x = w-2x+x^2$, where $w\ge0$, conserves energy

In physics courses we tend to use the following trick: multiply $\ddot x = w-2x+x^2$ by $\dot x$ to obtain $$ \dot x \ddot x = (w-2x+x^2)\dot x \ \ \Rightarrow \ \ \frac{d}{dt}\left( \frac{1}{2}\dot x^...
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10 votes

Functional equation $f(px)+p=[f(x)]^2$

This is a complete answer and goes a long way to explaining why the answer to this question is much more difficult than initially made out to be. Basically, there are such functions, and not only that ...
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10 votes
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Isn't my book wrongly equating $\frac{\frac{\sin^2x-\cos^2x}{\sin x\cos x}}{\frac{\sin^2x+\cos^2x}{\sin x\cos x}}$ and $-\cos2x$?

$\tan x $ is not defined when $\cos x =0$ and $\cot x $ is not defined when $\sin x =0$. So it is implicitly assumed the domain excludes points where $\sin x \cos x=0$.
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10 votes
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Why is the range a larger set than the domain?

Perhaps I'm understanding your question differently than some of the other commenters, but I'll point out that saying you have a function $g: D \to \mathbb{R}$ does NOT mean every element of $\mathbb{...
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10 votes
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Relation of Hamel basis with the equation $f(x + y) = f(x) + f(y)$?

The connection is the following one. A linear function from $\mathbb{R}$ to $\mathbb{R}$, where $\mathbb{R}$ is considered as a vector space over itself, is a function $f$ that satisfies the following ...
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10 votes
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Will $h(x)=\frac{ax+b}{cx+d}$, where $c\neq0$, always satisfy $h(h\cdots(h(x))\cdots)=x$ in a finite number of iterations?

is it always possible that the expression of form $h(x)$ = $\dfrac{ax+b}{cx+d}$ where $c \neq 0$ will satisfy $h(h\cdots (h(x))\cdots) = x$ Let's use the following notation: For a 2×2 matrix $$M=\...
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9 votes

Are there non-surjective functions in ZFC theory?

Surjectivity is a relationship between a function and a set. It is an extrinsic property of a function, when the function is defined as a set of ordered pairs. Not an intrinsic one like injectivity is....
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  • 373k
8 votes
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$f(x^2)=f(x)+f(-x)$

There are many solutions. We need $f(0)=0$. From there, define $f(x)$ however you like for positive values of $x$. Finally, for $x<0$, let $f(x) = f(x^2) - f(-x)$. Example: $$f(x) = \begin{cases}x^...
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8 votes
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If $\frac{f(x^2)}{f(x)}=1+x+x^2+\ldots+x^7$ then what is $f(x)$?

Simply use $$x^n+1=\frac{x^{2n}-1}{x^n-1}$$ for each factor of your decomposition $$(x+1)(x^2+1)(x^4+1).$$
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8 votes

Why isn't the Weierstrass function $\sum_{n=0}^\infty a^n \cos(b^n\pi x)$ differentiable?

While others have given answers saying that a pointwise limit of differentiable functions needn’t be differentiable, here is a simple example explaining why that is true. One can find a limit of ...
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8 votes
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In the category of Lie algebras are mono-/epimorphisms precisely the injective/surjective morphisms?

The other answer deals with monomorphisms, so let's look at epimorphisms. As pointed out in comments, there is a (1970?) preprint of G. Bergman's, Epimorphisms of Lie Algebras, available online, which ...
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8 votes
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Composing the empty function with itself

Notice how there really only is one function $\varnothing\to\varnothing$. Thus all your functions, i.e. $f$, $g$, $f\circ f$ and $\operatorname{id}_\varnothing$, are one and the same. The reason ...
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  • 3,501
8 votes

Does there exist a function which converts exponentiation into addition?

While your question as stated has been answered, I'd like to give a satisfying answer to a slight alteration of your question. To avoid the issue of commutivity, we can instead look at commutative ...
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  • 3,885
8 votes

Calculus: Difference between functions and "equations" from a theoretical perspective

It depends on the context what you mean with $$f(x,y) = x^2+y^2 \tag 1$$ What it could be is the definition of a function $f:\Bbb R^2\to\Bbb R$, or over any other domains where concepts like squaring ...
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8 votes

How do we know that mathematics is independent of the definition of the cartesian product?

One way of doing this is to introduce new notation $a, b \mapsto (a, b)$ and the axiom $(a, b) = (c, d) \to (a = c \land b = d)$ (as well as forms of replacement and separation which can have $(a, b)$ ...
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8 votes
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Does a function $f(x)$ exist such that $f(x+1)-f(x)=\frac{1}{x}$. And if so what is it.

Yes. Let $L(z)$ denote the logarithm of the gamma function. Then $L(x + 1) - L(x) = \log(x)$. So the derivative $L'(x)$ (i.e. $\Gamma'(x) / \Gamma(x)$) satisfies the desired functional equation.
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  • 25.7k
8 votes

Find the function $f(x)$ when $f(f(x))=1-x$, for $x\in [0,1]$

From $f(f(x)) =1-x$ follows $$1-f(x)=f(f(f(x)))=f(1-x) .$$ Hence, $$f(x) +f(1-x) =1$$ for all $x\in[0, 1]$.
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  • 2,032
7 votes
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Continuous surjective function from closed interval to itself that fix only the endpoints

You have already found the simple example $f(x)=x^2$ on the interval $[0,1]$. This example translates to every other interval $[a,b]$, where $a<b$. Simply take $$g(x)=(b-a)\left(\frac{x-a}{b-a}\...
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